Ultrasonic, flow measuring devices yield, besides data regarding the flow velocity of a medium and the velocity of sound of ultrasonic signals in the medium, also information concerning signal strengths and the signal-to-noise ratio. The evaluation of these variables and their tolerances is, however, left to the expert. Therefore, the usual plant technician responsible for servicing the plant and the device can only with difficulty estimate whether a failure of the measuring device is imminent. Known are ultrasonic, flow measuring devices, which display a critical state of the measuring system, for example, a signal strength that is too low. This occurs, however, only when there is no longer any signal at all and measurement has already stopped.